/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::Staircase

NAME

Math::PlanePath::Staircase -- integer points in stair-step diagonal stripes

SYNOPSIS

use Math::PlanePath::Staircase;
my $path = Math::PlanePath::Staircase->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path makes a staircase pattern down from the Y axis to the X,

 8      29
         |
 7      30---31
              |
 6      16   32---33
         |         |
 5      17---18   34---35
              |         |
 4       7   19---20   36---37
         |         |         |
 3       8--- 9   21---22   38---39
              |         |         |
 2       2   10---11   23---24   40...
         |         |         |
 1       3--- 4   12---13   25---26
              |         |         |
Y=0 ->   1    5--- 6   14---15   27---28

         ^
        X=0   1    2    3    4    5    6

The 1,6,15,28,etc along the X axis at the end of each run are the hexagonal numbers k*(2*k-1). The diagonal 3,10,21,36,etc up from X=0,Y=1 is the second hexagonal numbers k*(2*k+1), formed by extending the hexagonal numbers to negative k. The two together are the triangular numbers k*(k+1)/2.

Legendre's prime generating polynomial 2*k^2+29 bounces around for some low values then makes a steep diagonal upwards from X=19,Y=1, at a slope 3 up for 1 across, but only 2 of each 3 drawn.

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same pattern. For example to start at 0,

n_start => 0

28
29 30
15 31 32
16 17 33 34
 6 18 19 35 36
 7  8 20 21 37 38
 1  9 10 22 23 ....
 2  3 11 12 24 25
 0  4  5 13 14 26 27

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::Staircase->new ()
$path = Math::PlanePath::AztecDiamondRings->new (n_start => $n)

Create and return a new staircase path object.

$n = $path->xy_to_n ($x,$y)

Return the point number for coordinates $x,$y. $x and $y are rounded to the nearest integers, which has the effect of treating each point $n as a square of side 1, so the quadrant x>=-0.5, y>=-0.5 is covered.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.

FORMULAS

Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing pairs of N. Thus for rect_to_n_range() the lower left corner vertical pair is the minimum N and the upper right vertical pair is the maximum N.

A given X,Y is the larger of a vertical pair when ((X^Y)&1)==1. If that happens at the lower left corner then it's X,Y+1 which is the smaller N, as long as Y+1 is in the rectangle. Conversely at the top right if ((X^Y)&1)==0 then it's X,Y-1 which is the bigger N, again as long as Y-1 is in the rectangle.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

n_start=1 (the default)
  A084849    N on diagonal X=Y
  A210521    permutation N by diagonals, upwards
  A199855      inverse

n_start=0
  A014105    N on diagonal X=Y, second hexagonal numbers

n_start=2
  A128918    N on X axis, except initial 1,1
  A096376    N on diagonal X=Y

SEE ALSO

Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::Corner, Math::PlanePath::ToothpickSpiral

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde

This file is part of Math-PlanePath.

Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.